Collectionwise Hausdorff space

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In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.[1]

Here a subset being discrete has the usual meaning of being a discrete space with the subspace topology (i.e., all points of are isolated in ).[nb 1]

Properties

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  • Every collectionwise normal space is collectionwise Hausdorff. (This follows from the fact that given a closed discrete subset   of  , every singleton     is closed in   and the family of such singletons is a discrete family in  .)

Remarks

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  1. ^ If   is T1 space,   being closed and discrete is equivalent to the family of singletons   being a discrete family of subsets of   (in the sense that every point of   has a neighborhood that meets at most one set in the family). If   is not T1, the family of singletons being a discrete family is a weaker condition. For example, if   with the indiscrete topology,   is discrete but not closed, even though the corresponding family of singletons is a discrete family in  .

References

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